The problem: Given two random samples, s1 and s2, of size k over two disjoint populations, p1 and p2, how to combine the two k-sized random samples into one k-sized random sample over p1 ∪ p2? The solution: k times, draw an element s1 ∪ s2; with probability d1 = |p1| / |p1 ∪ p2|, …
Non-strict orders: ≤ The symbol ≤ denotes a generalization of “less than or equal”, and it defines either a partial or total ordering over a set P (in the table below a,b ∈ P): Constraint (Non-strict) partial order (Non-strict) total order Reflexivity: a ≤ a x x Antisymmetry: if a ≤ b and b ≤ …
Question: When can a distance function d(x,y) be called metric, pseudo-metric, quasi-metric or semi-metric? Constraint Metric Pseudo Quasi Semi Non-negativity: d(x,y) ≥ 0 x x x x Identity of indiscernibles: d(x,y)=0 ⇒ x=y x x x Symmetry: d(x,y) = d(y,x) x x x Triangle inequality: d(x,z) ≤ d(x,y)+d(y,z) x x x Table derived from Wikipedia …